Explanation

(a) When an atom comes down from some higher energy level to the first energy level then emitted lines form of Lyman series.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{\lambda_{L}}$ =$\ R\left( \frac{1}{1^{2}} - \frac{1}{n^{2}} \right)$
where$\ R$ is Rydberg’s constant.
When an atom comes from higher energy level to the second level, then Balmer series are obtained.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{\lambda_{B}}$ =$\ R\left( \frac{1}{2^{2}} - \frac{1}{n^{2}} \right)$
For maximum wavelength
$\ \ n$=2 , $\frac{1}{\lambda_{L}}$ =$\ R\left( 1 - \frac{1}{{(2)}^{2}} \right)$ = $R\left( 1 - \frac{1}{4} \right) = \frac{3R}{4}\ \ \ \ \ \ldots.\ (i)$
$n = 3$,$\frac{1}{\lambda_{B}}$ =$\ R\left( \frac{1}{{(2)}^{2}} - \frac{1}{{(3)}^{2}} \right)$ $= \ R\left( \frac{5}{36} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(ii)$
Dividing Eq. (ii) by Eq. (i), we get
$\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\ \lambda_{L}}{\lambda_{B}} = \frac{5}{27}$